Biquaternium

Gulielmus Rowan Hamilton eques.

BiquaterniumFormula:FD ref (Anglice biquaternion) in algebra abstracta est ullus ex numeris $w+xi+yj+zk$, ubi $w,x,y$, et $z$ sunt numeri complexi vel eorum variantes, et elementa $1,i,j,k$ se multiplicant ut in grege quaterniorum et cum eorum coefficientibus commutantur. Sunt tria biquaterniorum genera, quae cum numeris complexis eorumque variantibus congruunt:

• Biquaternia cum coefficientes numeri complexi sint
• Biquaternia fissa cum coefficientes numeri complexi et fissi sint
• Quaternia binaria cum coefficients numeri binarii sint.

Biquaternia ordinaria, a Gulielmo Rowan Hamilton excogitata, annis 1844 et 1850 in Proceedings of the Royal Irish Academy divulgata sunt. Inter gravissimos huius generis biquaterniorum suasores numerantur Arthurus W. Conway, Cornelius Lanczos, Alexander Macfarlane, Ludovicus Silberstein. Quasi-sphaerium, unitas biquaterniorum, gregem Lorentzianum repraesentat, qui fundamenta relativitatis specialis est.

Algebra biquaterniorum putari potest productus tensorum $ℂ\otimes ℍ$ (supra reales) ubi $ℂ$ est campus numerorum complexorum et $ℍ$ est algebra divisionis quaterniorum realium; quod breviter significat biquaternia esse complexificationem tantum quaterniorum. Biquaternia, cum genus algebrae complexae videantur, isomorphica ad algebram $2\times 2$ matricum complexarum $M_2(ℂ)$ sunt. Etiam sunt isomorphica ad nonnullas algebras Cliffordianas, inter quas $1=ℍ(ℂ)=C_3^0(ℂ)=C{\ell }_2(ℂ)=C{\ell }_{1,2}(ℝ)$, algebram Paulianam $C{\ell }_{3,0}(ℝ)$, et par algebrae spatiotemporalis pars $C{\ell }_{1,3}^0(ℝ)=C{\ell }_{3,1}^0(ℝ)$.

Bibliographia

• Buchheim, Arthur. 1885. "A Memoir on biquaternions." American Journal of Mathematics 7 (4): 293–326. JSTOR.
• Conway, Arthur W. 1911. "On the application of quaternions to some recent developments in electrical theory." Proceedings of the Royal Irish Academy 29A: 1–9.
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• Hamilton, William Rowan. 1866. Elements of Quaternions. Ed. a Gulielmo Eduino Hamilton, filio patris mortui. University of Dublin Press.
• Hamilton, William Rowan. 1899. Elements of Quaternions, vol. I, (1901). vol. II. Liber editus a Carolo Jasper Joly. Longmans, Green & Co.
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• Kravchenko, Vladislav. 2003. Applied Quaternionic Analysis. Heldermann Verlag. ISBN 3-88538-228-8.
• Sangwine, Stephen J., Todd A. Ell, et Nicolas Le Bihan. 2010. "Fundamental representations and algebraic properties of biquaternions or complexified quaternions." Advances in Applied Clifford Algebras 21 (3): 1–30. Formula:Doi. Arxiv 1001.0240.
• Sangwine, Stephen J., et Daniel Alfsmann. 2010. "Determination of the biquaternion divisors of zero, including idempotents and nilpotents." Advances in Applied Clifford Algebras 20 (2): 401–410. Bibcode 2008arXiv0812.1102S. Formula:Doi.
• Silberstein, Ludwik. 1912. "Quaternionic form of relativity." Philosophical Magazine series 6: 23 (137): 790–809. Formula:Doi. URL.
• Silberstein, Ludwik. 1914. The Theory of Relativity.
• Synge, J. L. 1972. "Quaternions, Lorentz transformations, and the Conway-Dirac-Eddington matrices." Communications of the Dublin Institute for Advanced Studies series A, vol. 21.
• Tanişli, M. 2006. "Gauge transformation and electromagnetism with biquaternions." Europhysics Letters 74 (4): 569. Bibcode = 2006EL.....74..569T. Formula:Doi.